{VERSION 5 0 "APPLE_PPC_MAC" "5.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 256 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 1 0 1 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Warning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 1 3 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 3" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 258 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 4 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 258 "" 0 "" {TEXT -1 18 "Mass-Spring System" }}{PARA 256 "" 0 "" {TEXT -1 10 "A. J. Meir" }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 257 "" 0 "" {TEXT -1 46 "Copyright (C) A. J. Meir. All rights r eserved." }}{PARA 257 "" 0 "" {TEXT -1 492 "This worksheet is for educ ational use only. No part of this publication may be reproduced or tra nsmitted for profit in any form or by any means, electronic or mechani cal, including photocopy, recording, or any information storage and re trieval system without prior written permission from the author. Not f or profit distribution of the software is allowed without prior writte n permission, providing that the worksheet is not modified in any way \+ and full credit to the author is acknowledged." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 4 "" 0 "" {TEXT -1 27 "Example: Mass-Spring Syst em" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 283 "An ordinary differential equation is an equation which involves an unkno wn function and its derivatives with respect to one independent variab le. (A partial differential equation involves an unknown function and \+ its partial derivatives with respect to several independent variables. )" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 89 "The \+ differential equation (initial value problem) describing a mass hangin g from a spring" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 259 "" 0 "" {XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d^2/(dt^2 );" "6#*&%\"dG\"\"#*$%#dtGF%!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "p; " "6#%\"pG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d/dt;" "6#*&%\"dG\"\"\"%#dtG!\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "p;" "6#%\"pG" }{TEXT -1 3 " + " }{XPPEDIT 18 0 "k; " "6#%\"kG" }{TEXT -1 1 " " }{XPPEDIT 18 0 "p;" "6#%\"pG" }{TEXT -1 13 " = F(t) " }{XPPEDIT 18 0 "p(0) = p[0];" "6#/-%\"pG6#\"\"!&F%6 #F'" }{TEXT -1 6 " " }{XPPEDIT 18 0 "d/dt;" "6#*&%\"dG\"\"\"%#dtG !\"\"" }{TEXT -1 1 " " }{XPPEDIT 18 0 "p(0) = p[1];" "6#/-%\"pG6#\"\"! &F%6#\"\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 "Here " } {XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 56 " is the mass of the mass (b ob) hanging from the spring, " }{XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 22 " the amount of damping" }{TEXT -1 7 "n, and " }{XPPEDIT 18 0 "k ;" "6#%\"kG" }{TEXT -1 47 " is the spring constant. The initial condit ion " }{XPPEDIT 18 0 "p[0];" "6#&%\"pG6#\"\"!" }{TEXT -1 29 " is the i nitial position and " }{XPPEDIT 18 0 "p[1];" "6#&%\"pG6#\"\"\"" } {TEXT -1 0 "" }{TEXT -1 38 "the initial velocity. Below we assume " } {XPPEDIT 18 0 "F(t) = 0;" "6#/-%\"FG6#%\"tG\"\"!" }{TEXT -1 1 "." }} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 0 "" }{TEXT 256 0 "" }{TEXT 257 14 "M aple commends" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plottools):with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 62 "Warning, the names arrow and changecoords have been redefined\n" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "oscillator:=proc(m,c,k,T,p 0,p1)" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "local eq, ic, soln, q, L, \+ C, N, seq1, seq2, h, spring, mass:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "global n, x, y, graph:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "eq:=m *diff(p(t),t,t)+c*diff(p(t),t)+k*p(t)=0:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "ic:=p(0)=p0,D(p)(0)=p1:" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "soln:=rhs(dsolve(\{eq,ic\},p(t))):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "graph:=plot(\{soln,diff(soln,t)\},t=0..T,thickness =2):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "q:=unapply(soln,t):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "L:=(x,y)->line([0,3],[x,y],thicknes s=3,color=red):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "C:=(x,y)->disk([ x,y],0.3,color=blue):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "N:=200:" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "x:=0:y:=-q((T/N)*n):" }}{PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 25 "seq1:=seq(L(x,y),n=0..N):" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 25 "seq2:=seq(C(x,y),n=0..N):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "h:=line([-4,3],[4,3],thickness=5,color=green):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "spring:=display(seq1,insequence=tru e):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "mass:=display(seq2,insequenc e=true):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "display(h,spring,mass,s caling=constrained):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 4 "end:" }}}} {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT -1 43 "We now prescribe values fo r the parameters " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "c;" "6#%\"cG" }{TEXT -1 6 ", and " }{XPPEDIT 18 0 "k;" "6#%\"kG" }{TEXT -1 157 ", give the final time (how long the simulatio n should run, note longer run times require more computer resources), \+ the initial position and initial velocity " }{XPPEDIT 18 0 "p[0];" "6# &%\"pG6#\"\"!" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "p[1];" "6#&%\"pG6# \"\"\"" }{TEXT -1 15 ", respectively." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "Mass:=1/4:Damping:=1/100:SpringConst:=2:Time:=20:\nIn itPosition:=1:InitVelocity:=0:" }}}{PARA 0 "" 0 "" {TEXT -1 110 "Below we obtain a simulation of the mass-spring system and also a plot of t he position and velosity of the bob" }}{PARA 0 "" 0 "" {TEXT -1 22 "as a function of time." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "osci llator(Mass,Damping,SpringConst,Time,InitPosition,InitVelocity);graph; " }}{PARA 13 "" 1 "" {GLPLOT2D 368 276 276 {PLOTDATA 2 "6$-%(ANIMATEG6 ew7%-%'CURVESG6%7$7$$\"\"!F-$\"\"$F-7$F,$!\"\"F--%'COLOURG6&%$RGBG$\"* ++++\"!\")F,F,-%*THICKNESSG6#F/-%)POLYGONSG6$7U7$$F/F2F17$$\"+/TMwH!#5 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